Revisiting Hua-Marcus-Bellman-Ando inequalities on contractive matrices
A complex \(n\times n\) matrix~\(M\) is strictly contractive if \(I-M^*M\) is positive definite. \textit{L.-K.~Hua} [Acta Math. Sinica~5, 463--470 (1955; Zbl 0066.26601), Am. Math. Soc. Transl., II. Ser.~32, 265--272 (1963; Zbl 0128.01704)] proved that if \(A\) and \(B\) are strictly contractive, then \[ I-B^*B+(A-B)^*(I-AA^*)^{-1}(A-B)= (I-B^*A)(I-A^*A)^{-1}(I-A^*B). \] He also proved certain determinantal inequalities. One of them is that if \(A_1,\dots,A_m\) are strictly contractive and \(k\geq n-1\), then the matrix~\(C=(c_{ij})\), \(c_{ij}=\det(I-A_i^*A_j)^{-k}\) is nonnegative definite. His work inspired further research by \textit{M.~Marcus} [Am. Math. Mon. 65, 266--269 (1958; Zbl 0083.00802)], \textit{R.~Bellman} [Duke Math.~J.~26, 485--490 (1959; Zbl 0087.01502)] and \textit{T.~Ando} [Linear and Multilinear Algebra~8, 347--352 (1980; Zbl 0438.15019)]. The present authors revisit these results with comments, analogs and extensions.
scientific article; zbMATH DE number 5518765
scientific article